(10 Pts) 8) A Cylindrical Water Tank Of Radius 10 Feet And Height 30 Feet Is Half Filled With Water. If
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Introduction
A cylindrical water tank with a radius of 10 feet and a height of 30 feet is a common structure used for storing large volumes of water. When such a tank is half filled with water, it presents interesting calculations related to volume, surface area, and other related parameters. Understanding these calculations is important for engineers, architects, and environmental planners to manage water resources efficiently. In this article, we will explore the various aspects of this cylindrical tank, including how to determine the volume of water when half filled, the surface area of the water level, and related practical considerations.
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Basic Dimensions and Volume of the Tank
Tank Dimensions
- Radius (r): 10 feet
- Height (h): 30 feet
Total Volume of the Tank
The total volume of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Calculating the total volume:
\[ V_{total} = \pi \times (10)^2 \times 30 \]
\[ V_{total} = \pi \times 100 \times 30 \]
\[ V_{total} = 3000 \pi \text{ cubic feet} \]
Using the approximation \( \pi \approx 3.1416 \):
\[ V_{total} \approx 3000 \times 3.1416 \]
\[ V_{total} \approx 9424.8 \text{ cubic feet} \]
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Water Volume When Half Filled
Since the tank is half filled with water, and the volume of a cylinder is proportional to its height, the volume of water is:
\[ V{water} = \frac{1}{2} \times V{total} \]
\[ V_{water} = \frac{1}{2} \times 3000 \pi \]
\[ V_{water} = 1500 \pi \text{ cubic feet} \]
Calculating:
\[ V_{water} \approx 1500 \times 3.1416 \]
\[ V_{water} \approx 4712.4 \text{ cubic feet} \]
Practical implication:
This means that when the tank is half filled, it contains approximately 4712.4 cubic feet of water.
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Determining the Water Level Height
In a cylindrical tank, the height of the water when half filled corresponds to half the total volume. Since the volume is proportional to height, the water height (h_water) when the tank is half filled is:
\[ h_{water} = \frac{h}{2} = \frac{30}{2} = 15 \text{ feet} \]
Therefore, the water level reaches a height of 15 feet.
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Surface Area of Water When Half Filled
Area of the Water Surface
The surface area of the water when half filled is the area of the circular cross-section at the water level:
\[ A = \pi r^2 \]
\[ A = \pi \times (10)^2 = 100 \pi \text{ square feet} \]
Numerical approximation:
\[ A \approx 100 \times 3.1416 = 314.16 \text{ sq ft} \]
This is the area of the water surface at the half-filled level.
Surface Area of the Water Surface
The surface area is significant for various reasons, such as calculating evaporation, surface water exposure, or designing access points.
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Surface Area of the Water's Curved Surface (Lateral Surface Area)
When the tank is half filled, it has a curved surface of water, which is the lateral surface area of the submerged part of the cylinder:
\[ \text{Lateral Surface Area} = 2 \pi r h_{water} \]
Where:
- \( r = 10 \) feet
- \( h_{water} = 15 \) feet
Calculating:
\[ \text{Lateral Surface Area} = 2 \pi \times 10 \times 15 = 300 \pi \]
Approximate value:
\[ 300 \times 3.1416 \approx 942.48 \text{ sq ft} \]
This is the surface area of the water’s curved side when the tank is half filled.
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Practical Applications and Considerations
- Water Storage Management
Understanding the volume when the tank is half filled helps in:
- Estimating the remaining capacity
- Planning refilling schedules
- Managing water supply during droughts or high demand
- Structural Design and Safety
Knowing the height of water at half capacity (15 feet) is crucial for:
- Ensuring the tank’s structural integrity under varying loads
- Designing overflow systems to prevent spillage
- Evaporation and Surface Area
The surface area at the water surface (314.16 sq ft) influences:
- Rate of water evaporation
- Effects of wind and temperature variations
- Pumping and Water Movement
Calculations of volume and surface area are essential for:
- Designing efficient pumping systems
- Estimating time for filling or draining the tank
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Additional Calculations and Related Concepts
A. Volume of Water at Different Fill Levels
If the tank is filled to a height \( h' \):
\[ V' = \pi r^2 h' \]
For example, at 10 feet:
\[ V' = \pi \times 100 \times 10 = 1000 \pi \approx 3141.6 \text{ cubic feet} \]
B. Surface Area at Different Fill Levels
The surface area of water at height \( h' \):
\[ A' = \pi r^2 = 100 \pi \approx 314.16 \text{ sq ft} \]
The area remains constant since the cross-sectional area of the cylinder is constant, regardless of water level.
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Summary of Key Points
- Total volume of the tank: approximately 9424.8 cubic feet
- Volume when half filled: approximately 4712.4 cubic feet
- Water height when half filled: 15 feet
- Surface area of water at half fill: approximately 314.16 square feet
- Lateral surface area of the water: approximately 942.48 square feet
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Conclusion
Understanding the geometric and volumetric properties of a cylindrical water tank is vital for efficient water resource management. When the tank is half filled, it contains about 4712.4 cubic feet of water, with a water surface area of approximately 314.16 square feet and a water height of 15 feet. These calculations aid in designing effective storage systems, ensuring structural safety, and optimizing water usage. Accurate knowledge of these parameters is essential for engineers, environmentalists, and facility managers involved in water storage and distribution.
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FAQs
Q1: How do the volume and surface area change if the tank is filled to a different level?
A: The volume varies linearly with height, calculated by \( V = \pi r^2 h' \). The surface area of the water’s top remains constant at \( 100 \pi \) sq ft, since it's the cross-sectional area at the water surface.
Q2: What is the significance of calculating the lateral surface area of water?
A: It helps in understanding the surface exposure of water, which impacts evaporation rates, surface temperature, and potential for surface-based processes.
Q3: How can these calculations assist in designing water systems?
A: They inform decisions related to pump capacities, overflow systems, structural load assessments, and water conservation strategies.
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By mastering these calculations, stakeholders can optimize water storage solutions, enhance safety, and promote sustainable water management practices.
